/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 9 According to Kepler's third law,... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 9

According to Kepler's third law, the orbital period \(T\) of a planet is related to the radius \(R\) of its orbit by \(T^{2} \propto R^{3}\) Jupiter's orbit is larger than Earth's by a factor of 5.19 What is Jupiter's orbital period? (Earth's orbital period is \(1 \text { yr. })\)

Short Answer

Expert verified
Answer: Jupiter's orbital period is approximately 11.86 years.

Step by step solution

01

Write the Kepler's third law formula

Kepler's third law states that the square of the period of two planets is proportional to the cube of their average distance from the sun. In mathematical form, the relationship becomes: $$\frac{T^{2}_{1}}{T^{2}_{2}} = \frac{R^{3}_{1}}{R^{3}_{2}}.$$
02

Assign values to the variables

According to the problem, Jupiter's orbit is larger than Earth's by a factor of 5.19, so we can write \(R_{2} = 5.19R_{1}\). Earth's orbital period (\(T_{1}\)) is 1 year, and we need to find Jupiter's orbital period (\(T_{2}\)).
03

Substitute the values into the equation and solve for \(T_2\)

Substitute the known values of \(R_{1}\), \(R_{2}\), and \(T_{1}\) into the equation: $$\frac{1^2}{T^{2}_{2}} = \frac{R^{3}_{1}}{(5.19R_{1})^{3}}.$$
04

Simplify the equation

Cubing 5.19 and simplifying the expression, we get: $$\frac{1}{T^{2}_{2}} = \frac{1}{5.19^{3}}.$$ Now, our objective is to find the value of \(T^{2}_{2}\).
05

Solve for \(T_2\)

To find \(T^{2}_{2}\), we can simply invert both sides of the equation: $$T^{2}_{2} = 5.19^{3}.$$ Now, to find the value of \(T_2\), we take the square root of both sides: $$T_{2} = \sqrt{5.19^{3}}.$$
06

Calculate Jupiter's orbital period

Using a calculator, we compute the value of \(T_2\): $$T_{2} 鈮� 11.86 \text{ years}.$$ Therefore, Jupiter's orbital period is approximately 11.86 years.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

The Space Shuttle astronauts use a massing chair to measure their mass. The chair is attached to a spring and is free to oscillate back and forth. The frequency of the oscillation is measured and that is used to calculate the total mass \(m\) attached to the spring. If the spring constant of the spring \(k\) is measured in \(\mathrm{kg} / \mathrm{s}^{2}\) and the chair's frequency \(f\) is \(0.50 \mathrm{s}^{-1}\) for a \(62-\mathrm{kg}\) astronaut, what is the chair's frequency for a 75 -kg astronaut? The chair itself has a mass of 10.0 kg. [Hint: Use dimensional analysis to find out how \(f\) depends on $m \text { and } k .]$
A sprinter can run at a top speed of 0.32 miles per minute. Express her speed in (a) \(\mathrm{m} / \mathrm{s}\) and (b) \(\mathrm{mi} / \mathrm{h}\).
The record blue whale in Problem 68 had a mass of $1.9 \times 10^{5} \mathrm{kg} .\( Assuming that its average density was \)0.85 \mathrm{g} / \mathrm{cm}^{3},$ as has been measured for other blue whales, what was the volume of the whale in cubic meters \(\left(\mathrm{m}^{3}\right) ?\) (Average density is the ratio of mass to volume.)
An object moving at constant speed \(v\) around a circle of radius \(r\) has an acceleration \(a\) directed toward the center of the circle. The SI unit of acceleration is \(\mathrm{m} / \mathrm{s}^{2} .\) (a) Use dimensional analysis to find \(a\) as a function of \(v\) and \(r\) (b) If the speed is increased $10.0 \%,$ by what percentage does the radial acceleration increase?
A spherical balloon expands when it is taken from the cold outdoors to the inside of a warm house. If its surface area increases \(16.0 \%,\) by what percentage does the radius of the balloon change?
See all solutions

Recommended explanations on Physics Textbooks

Electrostatics

Read Explanation

Conservation of Energy and Momentum

Read Explanation

Modern Physics

Read Explanation

Quantum Physics

Read Explanation

Physical Quantities And Units

Read Explanation

Kinematics Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.